Vol. 15, No. 3, October 2009
Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
numbers fromPr mes
Use building blocks to create a visual model for prime factorizations. Students can explore many concepts of number theory, including the relationship between greatest common factors and least common multiples.
Prime numbers are often described as the “building blocks” of natural numbers. This article will show how my students and I took this idea literally by using prime factorizations to build numbers with blocks. Many fascinating patterns and relationships emerge when a visual image of prime factorizations can be formed. This article will begin by exploring— 1. divisibility, 2. prime and composite numbers, and 3. properties of exponents. The article will conclude by investigating the relationship between— 1. greatest common factors and 2. least common multiples.
Using MUltiPlication to BUild coUnting nUMBeRs When we want to understand how something works in the physical world, we often look at how it is constructed from simpler pieces. If we want to know about the properties of molecules, we must understand that they are built from the elements that we see listed in the periodic table. Elements can then be analyzed by looking at their atomic structures. Similarly, when we understand that a number is built from its prime factors, we need to look at its properties and its relationship to other numbers. For the last few years, I have used the activities described here over a twoweek period in my sixth-grade classroom. Students come to sixth grade having been introduced to the basic definitions of prime and composite numbers and the procedures for finding prime factorizations. I place students in groups of two or three and distribute a set of colored centimeter cubes to each group; square “blocks” cut from colored paper or cardboard could also be used. The groups are told that each color represents a different number, although they do not know which particular number, and that placing Vol. 15, No. 3, October 2009
MatheMatics teaching in the Middle school
1 Fig. 1 a listing of prime numbers upFigure their representative color block to 20 and Color white red orange yellow green blue purple brown black* Prime Number Suggested Quantity per Group 20 10 5 5 3 3 2 2 5
a factor of 3. We need a new color of building block for the number 3, which is red. Students place a red block next to the number 3 on their page. Do we need a new type of building block to build 4? Students will see that although we could build 4 as 4 × 1, we may also use our known 2 blocks to express it as 2 × 2. Since we will never introduce a new type of building block unless it is needed, we attach two white 2 blocks to represent 2 × 2. Students place this pair of white blocks next to the number 4 on the page. As groups understand the process, they begin to work more independently. The next number is 5. Since it cannot be built using the factors of 2 or 3, the number 5 must have its own building block, which is orange. The number 6 can be written as 2 × 3. Students attach a white 2 block and a red 3 block. The number 7 requires a new block, which is yellow. Attaching three white 2 blocks as 2 × 2 × 2 represents 8. Students continue building the remaining counting numbers 2 through 12, as summarized in figure 2. Note that the commutative and associative properties of multiplication imply that a given collection of blocks represents a unique number, regardless of how they are ordered or grouped. The building blocks we have needed so far are 2, 3, 5, 7, and 11. What do these numbers have in common? Students often first point out that most of the...